The statistical properties of the volatility of price fluctuations

TL;DR

This paper investigates the statistical properties of market volatility, revealing power-law distributions, long-range correlations, and scaling behaviors in the S&P 500 and major companies over various time scales.

Contribution

It provides a comprehensive analysis of volatility distributions and correlations, applying novel methods like DFA to characterize market dynamics across different time horizons.

Findings

Volatility follows a power-law distribution with exponent ~4.

Long-range power-law correlations are observed in volatility.

Scaling behaviors are consistent across different time intervals.

Abstract

We study the statistical properties of volatility---a measure of how much the market is likely to fluctuate. We estimate the volatility by the local average of the absolute price changes. We analyze (a) the S&P 500 stock index for the 13-year period Jan 1984 to Dec 1996 and (b) the market capitalizations of the largest 500 companies registered in the Trades and Quotes data base, documenting all trades for all the securities listed in the three major stock exchanges in the US for the 2-year period Jan 1994 to Dec 1995. For the S&P 500 index, the probability density function of the volatility can be fit with a log-normal form in the center. However, the asymptotic behavior is better described by a power-law distribution characterized by an exponent 1 + \mu \approx 4. For individual companies, we find a power law asymptotic behavior of the probability distribution of volatility with exponent 1 + \mu \approx 4, similar to the S&P 500 index. In addition, we find that the volatility distribution scales for a range of time intervals. Further, we study the correlation function of the volatility and find power law decay with long persistence for the S&P 500 index and the individual companies with a crossover at approximately 1.5 days. To quantify the power-law correlations, we apply power spectrum analysis and a recently-developed modified root-mean-square analysis, termed detrended fluctuation analysis (DFA). For the S&P 500 index, DFA estimates for the exponents characterizing the power law correlations are \alpha_1=0.66 for short time scales (within \approx 1.5 days) and \alpha_2=0.93 for longer time scales (up to a year). For individual companies, we find \alpha_1=0.60 and \alpha_2=0.74, respectively. The power spectrum gives consistent estimates of the two power-law exponents.